R E P O R T. On Rejected Arguments and Implicit Conflicts: The Hidden Power of Argumentation Semantics INSTITUT FÜR INFORMATIONSSYSTEME

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1 TECHNICAL R E P O R T INSTITUT FÜR INFORMATIONSSYSTEME ABTEILUNG DATENBANKEN UND ARTIFICIAL INTELLIGENCE On Rejected Arguments and Implicit Conflicts: The Hidden Power of Argumentation Semantics DBAI-TR Institut für Informationssysteme Abteilung Datenbanken und Artificial Intelligence Technische Universität Wien Favoritenstr. 9 A-1040 Vienna, Austria Tel: Fax: sekret@dbai.tuwien.ac.at Ringo Baumann Wolfgang Dvořák Thomas Linsbichler Christof Spanring Hannes Strass Stefan Woltran DBAI TECHNICAL REPORT 2016

2 DBAI TECHNICAL REPORT DBAI TECHNICAL REPORT DBAI-TR , 2016 On Rejected Arguments and Implicit Conflicts: The Hidden Power of Argumentation Semantics Ringo Baumann 1 Wolfgang Dvořák 2 Thomas Linsbichler 3 Christof Spanring 4 Hannes Strass 5 Stefan Woltran 6 Abstract. Abstract argumentation frameworks (AFs) are one of the most studied formalisms in AI and are formally simple tools to model arguments and their conflicts. The evaluation of an AF yields extensions (with respect to a semantics) representing alternative acceptable sets of arguments. For many of the available semantics two effects can be observed: there exist arguments in the given AF that do not appear in any extension (rejected arguments); there exist pairs of arguments that do not occur jointly in any extension, albeit there is no explicit conflict between them in the given AF (implicit conflicts). In this paper, we investigate the question whether these situations are only a side-effect of particular AFs, or whether rejected arguments and implicit conflicts contribute to the expressiveness of the actual semantics. We do so by introducing two subclasses of AFs, namely compact and analytic frameworks. The former class contains AFs that do not contain rejected arguments with respect to a semantics at hand; AFs from the latter class are free of implicit conflicts for a given semantics. Frameworks that are contained in both classes would be natural candidates towards normal forms for AFs since they minimize the number of arguments on the one hand, and on the other hand maximize the information on conflicts, a fact that might help argumentation systems to evaluate AFs more efficiently. Our main results show that under stable, preferred, semi-stable, and stage semantics neither of the classes is able to capture the full expressive power of these semantics; we thus also refute a recent conjecture by Baumann et al. on implicit conflicts. Moreover, we give a detailed complexity analysis for the problem of deciding whether an AF is compact, resp. analytic. Finally, we also study the signature of these subclasses for the mentioned semantics and shed light on the question under which circumstances an arbitrary framework can be transformed into an equivalent compact, resp. analytic, AF.

3 1 Leipzig University, Germany. 2 University of Vienna, Austria. 3 TU Wien, Austria. 4 TU Wien, Austria and University of Liverpool, UK. 5 Leipzig University, Germany. 6 TU Wien, Austria. Acknowledgements: The authors are grateful to the anonymous reviewers of the preceding papers and the current article for their detailed reviews and helpful comments that led to an improved presentation. Furthermore, the authors are grateful to Paul Dunne for helpful discussions. This research has been supported by the German Research Foundation (DFG) under project BR 1817/7-1 and the Austrian Science Fund (FWF) under projects I1102, I2854, and P Copyright c 2016 by the authors 2

4 1 Introduction In recent years argumentation has emerged to become one of the major fields of research in Artificial Intelligence [34, 11]. In particular, Dung s well-studied abstract argumentation frameworks (AFs) [18] are a simple, yet powerful formalism for modeling and deciding argumentation problems that are integral to many advanced argumentation systems, see e.g. [12]. The evaluation of AFs in terms of finding reasonable positions with respect to a given framework is defined via so-called argumentation semantics (cf. [1] for an overview). Given an AF F, an argumentation semantics σ returns acceptable sets of arguments σ(f ), the (σ-)extensions of F. Several semantics have been introduced over the years [18, 39, 13, 2] with motivations ranging from the desired treatment of specific examples to fulfilling certain abstract principles. One important line of research in abstract argumentation is thus the systematic comparison of the different semantics available. Hereby, the behavior of extensions with respect to certain properties [4] has been analyzed and the expressive power of semantics [23, 26, 28, 36] has been studied by identifying the set of extension-sets achievable under certain semantics. On the other hand, subclasses of AFs such as acyclic, symmetric, odd-cycle-free or bipartite AFs, have been considered, since for some of these classes different semantics collapse [14, 19]. Beside these subclasses based on the graph structure there are also classes defined via properties of extensions. The probably most prominent such subclass is the class of coherent AFs [21], i.e. AFs where the stable and preferred semantics coincide. Further examples for subclasses that are defined via extensions can be found in [5, 25]. In this work we contribute to both streams of research. We introduce two new classes, which to the best of our knowledge have not received attention in the literature. The actual definition of these two classes is motivated by typical phenomena that can be observed for several semantics. First, there exist arguments in a given AF that do not appear in any extension. Since these so-called rejected arguments do not appear in the result of extension-based semantics, it is a natural question whether such arguments can be removed from the AF at hand without changing its outcome (in a certain way, this question is similar to the problem of simplifying propositional formulas by removing don t care atoms). In order to have a handle for analyzing the effect of rejected arguments, we introduce the class of compact AFs: an AF is compact (with respect to a semantics σ), if each of its arguments appears in at least one σ-extension. Second, we are interested in the concept of implicit conflicts. An attack between two arguments represents an explicit conflict. By the nature of most argumentation semantics, conflicts can however also be implicit in the sense that some arguments do not occur together in any extension, although there is no attack between them. In order to understand the expressive power of implicit conflicts we introduce the class of analytic frameworks. Given a semantics σ, if every conflict between two arguments a, b in an AF F is explicit (i.e., for all arguments a, b, if {a, b} E {a, b} for all σ-extensions E, then a attacks b in F or b attacks a in F ) then F is called analytic. Both compact and analytic AFs thus yield a semantic subclass since their definitions rely on the actual extensions obtained via the chosen semantics. The role of rejected arguments Although rejected arguments are natural ingredients in typical argumentation scenarios, it is of interest to understand in which ways rejected arguments contribute 2

5 Figure 1: Rejected argument x cannot be removed without changing the stable extensions. to the strength of a particular semantics. Let us first have a brief look on the naive semantics, which is defined as subset-maximal conflict-free sets: Here, it is rather easy to see that any AF can be transformed into an equivalent compact AF by just removing all self-attacking arguments. In other words, the same outcome (in terms of the naive extensions) can be achieved by a simplified AF without rejected arguments. On the one hand, this can be seen as a general weakness of naive semantics, since any possible outcome can be equivalently achieved in the absence of rejected arguments. On the other hand, this shows that towards evaluating an AF under naive semantics, the transformation into a compact AF can provide a beneficial pre-processing step for computing the extensions (which afterwards should however be interpreted in terms of the original AF). How is the situation with semantics that are considered more mature? We borrow an example from (author?) [22]. Consider the AF F 1 in Figure 1, where nodes represent arguments and directed edges represent attacks. The stable extensions (conflict-free sets attacking all other arguments) of F 1 are given by the set S = {{a, b, c}, {a, b, c }, {a, b, c }, {a, b, c}, {a, b, c }, {a, b, c}, {a, b, c}}. Observe that x is rejected, i.e. x does not appear in any stable extension of F 1. Hence, this framework is not compact for the stable semantics. Moreover, it was shown in [22] that there is no compact AF (in this case an AF not using argument x) that yields the same stable extensions as F 1. By the necessity of conflict-freeness any such compact AF would only allow conflicts between arguments a and a, b and b, and c and c, respectively. Moreover, there would have to be attacks in both directions for each of these conflicts in order to ensure stability. Hence any compact AF having the same stable extensions as F 1 necessarily yields {a, b, c } in addition. In other words, under the stable semantics particular outcomes (in the example the set S of extensions) can only be achieved via AFs containing at least one rejected argument. Thus, the stable semantics makes proper use of rejected arguments. As we will see, all semantics under consideration (except naive semantics) show a similar behaviour. The role of implicit conflicts As introduced earlier, implicit conflicts arise when two arguments are never jointly accepted although they do not attack each other. The AF F 2 in Figure 2 provides a simple example for this effect. It can be seen that stable semantics yields two extensions {a, d} and {b, c} for F 2. Since c and d do not occur together in an extension there is an implicit conflict and thus F 2 is not analytic (for stable semantics). However, the naive extensions of F 2 are given by {a, d}, {b, c}, {c, d}. Thus c and d are not in an implicit conflict here, and the AF is easily seen to be analytic for naive semantics. 3

6 Figure 2: AF illustrating an implicit conflict between c and d for stable semantics. Indeed, by definition of naive semantics, two arguments occur together in a naive extension if and only if there is no attack between them and they are not self-attacking. Thus not every AF is analytic for naive semantics, but it is quite easy to see that every AF can be turned into an equivalent analytic one over the same arguments, by just connecting the self-attacking arguments to any other argument. Coming back to our example and to stable semantics, the question remains whether F 2 can be turned into an equivalent analytic one? This is quite an easy exercise. Just add an attack from c to d, or likewise from d to c. In fact, this addition does not change the set of extensions. However, it has been left as open question in [7] (stated as Explicit Conflict Conjecture ) whether such a manipulation of an AF is always possible. In this work, we shall negatively answer this question showing that (i) for preferred and semi-stable semantics, there exist AFs such that there is no equivalent analytic AF; and (ii) for stable and stage semantics, there exist AFs such that there is no equivalent analytic AF, unless we are allowed to add rejected arguments. Expressiveness of compact and analytic argumentation frameworks Before giving an overview of the obtained results, let us further illustrate some issues that come along with the subclasses of compact and analytic argumentation frameworks. One natural question is whether any AF F can be transformed to an equivalent AF G, i.e. σ(f ) = σ(g) for a given semantics σ, that is compact or analytic. In case the answer is no, we can conclude that the full range of expressiveness of σ indeed relies on the concepts of rejected arguments and implicit conflicts. Knowing which sets of extensions a semantics is able to express is of central interest in approaches of extension-based revision of AFs [16]. As the result of the revision may also be subject to certain syntactic constraints (e.g. a fixed set of arguments [15]) it is important to know about the role of rejected arguments and implicit conflicts. For instance, a revised AF might be required (e.g. in order to fulfill revision postulates) to have exactly the extension-set S from above under stable semantics while consisting solely of the arguments {a, b, c, a, b, c }. As we have already observed, and we will show in a more comprehensive and general manner in the paper, such a revision is not possible since getting S under stable semantics would require an additional, rejected argument. Implications for argumentation systems An even more promising application of our results lies in the field of concrete software systems for computing semantics of abstract argumentation frameworks. A considerable number of such systems ( solvers ) exist, as has been witnessed by the First International Competition on Computational Models of Argument (ICCMA 2015) [38]. 1 Using instances from that competition and additional instances created according to the same graph model as the competition instances, we also performed an experimental evaluation on the theoret- 1 A total number of 18 solvers participated, see 4

7 ical phenomena we study in this paper. The results can be found in A, and demonstrate the clear computational benefit of knowing about implicit conflicts in an argumentation framework. More precisely, once all implicit conflicts of an AF are made explicit, then the competition winners are able to compute the AF s extensions (for stable and preferred semantics) much faster than before (without implicit conflicts made explicit). Thus it is a naturally arising research question whether information about implicit conflicts can be obtained cheaply in terms of computational cost, a question that we will also address in the paper. For knowing about rejected arguments, the computational gain is immediately clear, since the lower the number of arguments, the smaller is the search space a solver has to go through in order to find all extensions. Thus, preprocessing steps that remove rejected arguments might also be beneficial to solving runtime. Moreover, if an AF has no rejected arguments then all of its arguments are contained in at least one extension, and so credulous as well as skeptical reasoning become easy tasks [7]. Overall, the research question we are interested in is: how computationally costly is it to determine whether an AF can be simplified along the dimensions rejected arguments and implicit conflicts? Answering this question would be crucial towards the development of clever methods for preprocessing AFs before solving. However, more fundamental questions need to be addressed first. On the one hand, we analyse how hard it is to decide whether an AF is compact (resp. analytic); on the other hand, we ask whether any AF can be transformed into an AF that is compact (resp. analytic) and equivalent under a particular semantics. Unfortunately, the answers to both of these questions is in a certain sense negative for all of the semantics we consider: intuitively speaking, our complexity results will show that deciding whether simplification is applicable (having certain reasoning tasks in mind) is as expensive as solving the reasoning tasks themselves. Furthermore, we can even show that there are AFs that cannot be exhaustively simplified. (More formally, there are AFs that have pathological implicit conflicts that cannot be made explicit even if we allow arbitrary semantics-preserving changes in other parts of the AF.) This does not make our results less applicable to implementation of reasoning systems, however. These negative results help the solver development community to delineate what can and cannot be done in improving solver performance by intelligent preprocessing. That is, by our results, we know that computing all rejected arguments and implicit conflicts are not viable candidates for simplifying given argumentation frameworks. Main contributions & structure of the paper The main contributions of this article are organized as follows. Recall that the semantics we mainly investigate are stable, preferred, semi-stable, stage, and naive semantics. In Section 3 we formally introduce the subclasses of compact and analytic AFs with respect to the considered semantics and investigate their relationship. For both classes the picture is similar: for instance, if an AF is compact (resp. analytic) for stable it also is for semi-stable (preferred, stage, and naive); but the other direction does not hold in general. Section 4 answers the question how hard it is to decide whether an AF is compact (resp. analytic). As it turns out, the complexity of this problem for a given semantics σ is the 5

8 same as credulous acceptance under σ. Thus, we obtain tractability for naive semantics, NPcompleteness for stable and preferred semantics, and Σ P 2 -completeness for semi-stable and stage semantics. In Section 5 we refute the Explicit Conflict Conjecture [7] for σ being among stable, preferred, semi-stable and stage semantics. In fact, we provide AFs such that there is no AF equivalent under σ that contains solely explicit conflicts. On the other hand, we identify sufficient conditions guaranteeing equivalence-preserving translations to analytic AFs. The final collection of results in Section 6 is concerned with signatures for compact and analytic frameworks. Signatures as introduced in [23] plainly collect all possible sets of extensions AFs can deliver under a given semantics. For instance, it is shown in [23] that preferred and semi-stable semantics yield an equal signature Σ, while the signature of stage semantics is a proper subset of Σ. Compared to [23], we do not give exact characterizations of signatures for compact (resp. analytic) frameworks, but obtain a full picture of their relationship with respect to the different semantics. For instance, we show that in terms of compact AFs, the signatures for semi-stable and preferred semantics become incomparable, while for analytic AFs, the signature for semi-stable semantics is a proper subset of the signature for preferred semantics. Finally, we generalize some recent results on maximal numbers of extensions [9] to give some impossibility results for compact realizability. In this work we consider several rather complex examples of argumentation frameworks, whose evaluation is a non-trivial task. Thus, for the reader s convenience, we provide encodings in the.apx format, which can be used to evaluate the AFs with systems like ASPARTIX [29] 2. These encodings can be either downloaded from HiddenPowerAFs.zip or directly accessed by clicking at the corresponding figure. (Depending on the actual pdf viewer, a right or double-click should initiate saving.) This article is based on [7] and [32], but also contains several new results. 2 Preliminaries In what follows, we briefly recall the necessary background on abstract argumentation and computational complexity. For an excellent overview on abstract argumentation and in particular on argumentation semantics, we refer to [1]. Abstract Argumentation Throughout the paper we assume a countably infinite domain A of arguments. An argumentation framework (AF) is a pair F = (A, R) where A A is a finite set of arguments and R A A is the attack relation. The collection of all AFs is given as AF A. For an AF F = (B, S) we use A F and R F to refer to B and S, respectively. We write a F b for (a, b) R F and S F a 2 A web front-end is available at 6

9 (resp. a F S) if there exists some s S such that s F a (resp. a F s). Symmetric attacks {(a, b), (b, a)} R F are occasionally denoted by a, b R F. For S A, the range of S (w.r.t. F ), denoted S + F, is the set S {b S F b}. Moreover, F S denotes the AF (A F S, R (S S)). Given an AF F, an argument a A F is defended (in F ) by S A F if for each b A F, such that b F a, also S F b. A set T of arguments is defended (in F ) by S if each a T is defended by S (in F ). A set S A F is conflict-free (in F ), if there are no arguments a, b S, such that (a, b) R F. We denote the set of all conflict-free sets in F as cf(f ). S cf(f ) is called admissible (in F ) if S defends itself. We denote the set of admissible sets in F as adm(f ). The terms semantics and extension are often used almost synonymously. Formally a semantics is a mapping, while extensions are concrete elements of its image. The semantics we study in this work are those characterized by the naive, stable, preferred, stage, and semi-stable extensions. Given an AF F they are defined as subsets of cf(f ) as follows: S nai(f ), if T cf(f ) with T S; S stb(f ), if S + F = A F ; S prf(f ), if S adm(f ) and T adm(f ) with T S; S stg(f ), if T cf(f ) with T + F S+ F ; S sem(f ), if S adm(f ) and T adm(f ) with T + F S+ F. The following relations between these semantics are well-known to hold for any AF F : stb(f ) sem(f ) prf(f ) stb(f ) stg(f ) nai(f ) Furthermore, apart from stable semantics all considered semantics guarantee the existence of at least one (possibly empty) extension as long as finite AFs are considered (cf. [8] for a detailed overview including the infinite case). We will also make frequent use of the following concepts. Definition 1. Given S 2 A, Args S denotes S S S and Pairs S denotes {(a, b) S S : {a, b} S}. S is called an extension-set (over A) if Args S is finite. In words, Args S stands for all arguments occurring in some element of S and Pairs S for all pairs of arguments occurring together in some element of S. As is easily observed, for all semantics σ, σ(f ) is an extension-set for any AF F. Example 1. Consider the AF F depicted in Figure 3. We have that a, b and f are self-attacking, since all semantics considered build upon conflict-freeness these three arguments can thus not be included in any extension. Similarly we may accept only one argument from a and c as these two are mutually attacking each other; the same holds for b and d and also for c and d. Naive semantics generates maximal conflict-free sets, we thus get nai(f ) = {{a, b, e}, {a, d, e}, {b, c, e}}. 7

10 Figure 3: Argumentation framework F used in Example 1. Argument e is contained in each naive extension as it does not share any attacks with not selfattacking arguments. However e is attacked by f and can not defend itself against this attack. Thus the set {a, b, e} is not admissible. Preferred semantics, as maximal admissible sets, then computes to prf(f ) = {{a, b}, {a, d, e}, {b, c, e}}. Now for stable extensions we need conflict-freeness as well as a partition of all arguments into accepted or attacked. Naturally this means that only maximal conflict-free sets are candidates. However neither of the naive sets has full range: {a, b, e} does not attack f, {a, d, e} does not attack b and {b, c, e} does not attack a. Thus there is no stable extension, i.e. stb(f ) =. Stage semantics can be seen as a less restrictive form of stable semantics in that we do not need to cover all arguments in range but want extensions to be conflict-free and range-maximal. As emphasized above all naive extensions have incomparable range (missing f, b, or resp. a ) and thus stg(f ) = nai(f ). Similarly semi-stable extensions are those admissible sets that are rangemaximal. And in this case also the preferred extensions all have incomparable range (missing f and e, b, or resp. a ) and thus sem(f ) = prf(f ). Now as for the concepts introduced in Definition 1 we have Args which are all the arguments occurring in any extension; in this case for all semantics σ {nai, stg, sem, prf} we get Args σ(f ) = {a, b, c, d, e}. And we have Pairs, all pairs of arguments that occur together in any extension; in this case as can easily be checked again for all semantics σ {naive, stg, sem, prf} we get Pairs σ(f ) = {(a, b), (b, a), (a, e), (e, a), (b, e), (e, b), (a, d), (d, a), (d, e), (e, d), (b, c), (c, b), (c, e), (e, c), (a, a), (b, b), (c, c), (d, d), (e, e)}. Computational Complexity We assume the reader is familiar with standard complexity concepts, such as P, NP and completeness. Nevertheless we briefly recapitulate the concept of NP-oracle machines and the related complexity class Σ P 2. By an NP-oracle machine we mean a Turing machine that can access an oracle that decides a given sub-problem from NP (or conp) within one step. The class Σ P 2 (sometimes also denoted by NP NP ), contains the problems that can be decided in polynomial time by a nondeterministic NP-oracle machine. The known complexity results for the argumentation semantics under consideration are summarized in Table 1 [13, 17, 19, 21, 27]. Here, Ver σ refers to the problem of verifying that a given set is an extension of a given arbitrary AF F w.r.t. the semantics σ; Cred σ refers to the problem of verifying that a given argument x is credulously accepted w.r.t. σ in F (there is at least one σ- 8

11 Table 1: Complexity of decision problems (C-c denotes completeness for class C). Ver σ Cred σ Skept σ nai in P in P in P stb in P NP-c conp-c adm in P NP-c trivial prf conp-c NP-c Π P 2 -c stg conp-c Σ P 2 -c Π P 2 -c sem conp-c Σ P 2 -c Π P 2 -c extension of F containing x); and Skept σ refers to the problem of verifying that a given argument x is skeptically accepted w.r.t. σ in F (x is contained in each σ-extension of F ). For a more detailed discussion of the complexity results the interested reader is referred to [20, 24]. We only mention that the hardness results still hold if restricted to frameworks without self-attacking arguments, which we will make use of later on. Later, for semantics σ, we will also need upper bounds for the problem Cred 2 σ defined as follows: given AF F and arguments a and b, does there exist an extension S σ(f ) such that {a, b} S (see e.g. [19]). For the semantics under consideration, it is rather straightforward to see that membership for Cred σ carries over to Credσ. 2 For σ {prf, stb, sem, stg} this is witnessed by the standard NP-algorithm of guessing a set S containing a and b and apply an oracle for verifying whether S is a σ-extension. The complexity of the verification problem then yields the desired upper bound. Membership in P for the naive semantics can be decided by just checking whether a, b are neither self-attacking nor attacking each other. Indeed, in this case {a, b} is conflict-free in the given AF F, and thus there must exist a naive extension of F containing both a and b. 3 Subclasses of Argumentation Frameworks In this section, we formally introduce the two central subclasses of argumentation frameworks of this paper, namely compact and analytic frameworks. We study basic properties and relations within the classes first. At the end of the section we will compare the two classes. 3.1 Compact Argumentation Frameworks The main idea behind compact argumentation frameworks is the absence of rejected arguments (w.r.t. a given semantics). Definition 2. Given a semantics σ, an AF F is called compact for σ (or σ-compact) if Args σ(f ) = A F. The set of all compact argumentation frameworks for σ is denoted by CAF σ. 9

12 Figure 4: AF discussed in Example 2, which is prf-compact but neither sem-compact nor stgcompact. Example 2. Let us consider the AF F depicted in Figure 4. 3 The preferred extensions of F are prf(f ) = {{z}, {x 1, a 1 }, {x 2, a 2 }, {x 3, a 3 }, {y 1, b 1 }, {y 2, b 2 }, {y 3, b 3 }}, meaning that F is prfcompact (F CAF prf ) since each argument occurs in at least one preferred extension. On the other hand observe that sem(f ) = prf(f )\{{z}} and stg(f ) = {{x i, a i, b j }, {y i, b i, a j } 1 i, j 3}, i.e. z is not contained in any semi-stable or stage extension. Therefore F is neither compact for semi-stable nor compact for stage semantics (i.e. F / CAF sem and F / CAF stg ). As indicated by Example 2, the contents of CAF σ differ with respect to the semantics σ. Concerning relations between the classes of compact AFs we start with an easy observation. In the following result, the only requirement on a semantics σ is that extensions are subsets of the arguments in the framework, i.e. Args σ(f ) A F for any AF F. Lemma 1. For any two semantics σ and θ such that for each AF F, for every S σ(f ) there is some S θ(f ) with S S, we have CAF σ CAF θ. Proof. Suppose F CAF σ. By definition, Args σ(f ) = A F. Now if for each S σ(f ) there is some S θ(f ) with S S, we have Args σ(f ) Args θ(f ). Since Args θ(f ) A F by definition, Args θ(f ) = A F follows. Hence, F CAF θ. Note that the case where σ(f ) θ(f ) holds for each AF F is a special case of the premise of Lemma 1. The next result provides a full picture of the relations between classes of compact AFs for the semantics we consider (see also Figure 5). Theorem 2. The following relations hold: 1. CAF stb CAF σ CAF nai for σ {prf, sem, stg}; 2. CAF sem CAF prf ; 3. CAF stg CAF θ and CAF θ CAF stg for θ {prf, sem}. Proof. (1) Let σ {prf, sem, stg}. The -relations are due to Lemma 1 together with following facts: (a) in any AF F, stb(f ) σ(f ); (b) each σ-extension E of an AF F is conflict-free in F, thus there exists a naive extension E of F with E E. 10

13 CAF σ CAF nai : The AF ({a, b}, {(a, b)}) is compact for naive semantics but not for σ. CAF stb CAF σ : Consider AF F from Figure 6a. We have prf(f ) = sem(f ) = {{x 1, a 1 }, {x 2, a 2 }, {x 3, a 3 }, {y 1, b 1 }, {y 2, b 2 }, {y 3, b 3 }}, and each of these extensions can be extended to a stage extension (the former three by adding one of the arguments b 1, b 2, b 3 the latter three by adding one of the arguments a 1, a 2, a 3 ), but stb(f ) =. Thus A F = Args σ(f ) Args stb(f ) =, meaning that F CAF σ but F / CAF stb. (2) CAF sem CAF prf is by the fact that, in any AF F, sem(f ) prf(f ) (cf. Lemma 1). Properness is by the AF in Figure 4, which is (as discussed in Example 2) prf-compact but not sem-compact. (3) First we show CAF stg CAF θ for θ {prf, sem}. To this end, consider the simple AF F = ({a, b, c}, {(a, b), (b, c), (c, a)}). We have stg(f ) = {{a}, {b}, {c}}, thus F CAF stg. On the other hand, sem(f ) = prf(f ) = { }, thus F / CAF σ. CAF prf CAF stg follows by the observations in Example 2. CAF sem CAF stg : Consider the AF F in Figure 6b. One can check that this AF is semcompact, but not stg-compact. In fact, argument a does not occur in any stage extension. Although {a, u 1, x 5 }, {a, u 2, x 6 }, {a, u 3, x 7 } sem(f ), the range of any conflict-free set containing a is a proper subset of the range of every stage extension of F : stg(f ) = {{c, u i, x 4 } i {1, 2, 3}} {{b, u i, s j, x i+4 } i, j {1, 2, 3}} {{t i, u j, s i, x i } i, j {1, 2, 3}}. Hence CAF sem CAF stg. Finally note that every symmetric and irreflexive (i.e. no self-attacking arguments) AF is contained in CAF stb, as already observed in [14, Proposition 6], and therefore also in each CAF σ for all semantics σ under consideration. But already CAF stb contains strictly more AFs than the class of symmetric and irreflexive AFs, which is, for instance, indicated by the AF ({a, b, c, d}, {(a, b), (b, c), (c, d), (d, a)}), which is clearly not symmetric but compact for the stable semantics. On the other hand observe that CAF nai AF A, as every AF having self-attacking arguments is not contained in CAF nai. 3.2 Analytic Argumentation Frameworks In this section we deal with AFs containing no implicit conflicts, which we will call analytic. We differentiate between the concept of an attack (as a syntactical element) and the concept of a conflict (with respect to the evaluation under a given semantics). 3 The construct in the lower part of the figure represents symmetric attacks between each pair of distinct arguments. We will make use of this style in illustrations throughout the paper. CAF stb CAF sem CAF stg CAF prf CAF nai Figure 5: Relations between classes of compact AFs (cf. Theorem 2). 11

14 (a) AF F contained in CAF prf, CAF sem, and CAF stg but not in CAF stb. (b) AF F contained in CAF sem but not in CAF stg. Figure 6: AF used in the proof of Theorem 2 to show the incomparability of certain classes of compact AFs. Definition 3. Given some AF F = (A, R), a semantics σ and arguments a, b A. If (a, b) / Pairs σ(f ), we say that a and b are in conflict in F for σ. If (a, b) R or (b, a) R we say that the conflict between a and b is explicit, otherwise the conflict is called implicit (with respect to σ). Notice that Definition 3 does not require a and b to be different arguments. In particular, an argument that is not contained in any σ-extension is in conflict with itself. This conflict is explicit if the argument is self-attacking and implicit otherwise. Definition 4. Given a semantics σ, an AF F is called analytic for σ (or σ-analytic) if all conflicts of F for σ are explicit in F. The set of all analytic argumentation frameworks for σ is denoted by XAF σ. Example 3. As a simple example consider the AF F 2 from the introduction, depicted in Figure 2. For σ {stb, prf, sem, stg} we have σ(f 2 ) = {{a, d}, {b, c}}. Observe that there is an implicit conflict between arguments c and d, denoted by a dashed line in Figure 2. Hence F 2 is not σ- analytic, i.e. F 2 / XAF σ. Observe however that nai(f 2 ) = σ(f 2 ) {{c, d}}, which means that F 2 is analytic for naive semantics. As indicated in Example 3 the sets of analytic AFs can differ for different semantics. Again, well-known relations between the extensions of certain semantics allow us to obtain -relations between classes of analytic AFs. Lemma 3. For any two semantics σ and θ such that for each AF F, for every S σ(f ) there is some S θ(f ) with S S, we have XAF σ XAF θ. 12

15 XAF stb XAF sem XAF stg XAF prf XAF nai Figure 7: Relations between classes of analytic AFs (cf. Theorem 4). Proof. Let F XAF σ and let there be a conflict between arguments a, b A F for θ, i.e. (a, b) / Pairs θ(f ). Now since for every S σ(f ) there is some S θ(f ) with S S it follows that Pairs σ(f ) Pairs θ(f ). Hence also (a, b) / Pairs σ(f ). By the assumption that F XAF σ we know that there is an attack a F b or b F a, hence also F XAF θ. Similarly as for compact AFs, observe that every symmetric and irreflexive (i.e. no selfattacking arguments) AF is contained in XAF σ for all semantics under consideration. The next result provides a full picture of the relations between classes of analytic AFs for the semantics we consider (see also Figure 7). We will frequently use Lemma 3, with either the exact condition or the special case σ(f ) θ(f ). Theorem 4. The following relations hold: 1. XAF stb XAF σ XAF nai for σ {prf, sem, stg}; 2. XAF sem XAF prf ; 3. XAF stg XAF θ and XAF θ XAF stg for θ {prf, sem}. Proof. (1) Let σ {prf, sem, stg}. The -relations are due to Lemma 3 together with following facts: (a) in any AF F, stb(f ) σ(f ); (b) each σ-extension E of an AF F is conflict-free in F, thus there exists a naive extension E of F with E E. XAF σ XAF nai : The AF in Figure 2 is, as discussed in Example 3, nai-analytic but not σ- analytic. XAF stb XAF σ : Consider the AF F from Figure 8. It contains several kinds of complete subframeworks, in the sense that each member of such a subframework attacks each other member. Two complete subframeworks of nine arguments ({r i, u i, x i 1 i 3} and {s i, v i, y i 1 i 3}) and three complete subframeworks of six arguments ({r i, s i 1 i 3}, {u i, v i 1 i 3} and {x i, y i 1 i 3}). Further there are three directed three-cycles (among {a i 1 i 3}, {b i 1 i 3} and {c i 1 i 3}), and each argument from the complete subframeworks attacks exactly two arguments from one three-cycle, effectively activating the third one. Now observe that we have stb(f ) =, as at least one argument of a i, b i, c i remains out of range due to conflict-freeness, i.e. a conflict-free set in F can have only one argument from each complete nine-component and thus leaves at least one of the three-cycles unattacked. Therefore there is an implicit conflict for stb for every pair of non-attacking arguments, hence F / XAF stb. On the other hand we have prf(f ) = sem(f ) = {{r i, v j, a i, b j }, {s i, u j, a i, b j }, {r i, y j, a i, c j }, {s i, x j, a i, c j }, {u i, y j, b i, c j }, {v i, x j, b i, c j } 1 i, j 3} and stg(f ) = {{r i, v j, a i, b j, c k }, {s i, u j, a i, b j, c k }, 13

16 Figure 8: AF F with F XAF σ for σ {prf, sem, stg} and F XAF stb. Figure 9: AF F with F XAF prf and F XAF σ for σ {stb, sem, stg}. {r i, y j, a i, c j, b k }, {s i, x j, a i, c j, b k }, {u i, y j, b i, c j, a k }, {v i, x j, b i, c j, a k } 1 i, j, k 3}, which allows to verify that all conflicts for σ are explicit in F, hence F XAF σ. (2) By Lemma 3 we get XAF sem XAF prf. In order to obtain properness of this relation consider the AF F from Figure 9 and define a cyclic successor function s as s(1) = 2, s(2) = 3, s(3) = 1, and s(4) = 5, s(5) = 6, s(6) = 4. We have sem(f ) = {{x i, y j, z s(i), z s(j) } i {1, 2, 3}, j {4, 5, 6} or i {4, 5, 6}, j {1, 2, 3}}, yielding plenty of implicit conflicts, e.g. between x i and y i. Hence F is not analytic for semi-stable semantics. We further define s({i}) = s(i) and for s(i) = j also s({i, j}) = s(j). Now on the other hand we have prf(f ) = sem(f ) {{x i, y j, z s({i,j}) } i, j {1, 2, 3} or i, j {4, 5, 6}}, witnessing F XAF prf. (3) XAF stg XAF σ : Consider a directed cycle of five arguments F, A F = {x 1, x 2, x 3, x 4, x 5 } and R F = {(x 1, x 2 ), (x 2, x 3 ), (x 3, x 4 ), (x 4, x 5 ), (x 5, x 1 )}. Here we have stg(f ) = {{x 1, x 3 }, {x 1, x 4 }, {x 2, x 4 }, {x 2, x 5 }, {x 3, x 5 }} and thus F XAF stg. On the other hand sem(f ) = prf(f ) = { }, marking all pairs of arguments as being in conflict and thus for instance the conflict between x 1 and x 3 is implicit for prf and sem (and also stb). XAF prf XAF stg : The AF F in Figure 9 is, as argued in (2), explicit for prf, but not for sem. 14

17 Figure 10: AF F with F XAF sem for F XAF stg. Here F X refers to the AF from Figure 8 and x is in a symmetric attack relationship with all arguments from F X. However, it holds that stg(f ) = sem(f ), hence also F / XAF stg. XAF sem XAF stg : As witness of XAF sem XAF stg consider the AF F from Figure 10. This AF is composed of two subframeworks, F X from Figure 8 and F C from Figure 6b, and a connecting interface consisting of argument x and its counterpart set Y = { s i, t i, ū i i {1, 2, 3}}. There are symmetric attacks between the members ᾱ of Y and their counterparts α from F C, between x and all members of Y, and between x and all arguments from F X. A key ingredient to this construction is that both, F C and F X, on their own do not provide stable extensions and thus at least one argument remains out of range for any stage or semi-stable extension. In addition observe that F X is compact for both semi-stable and stage, while F C is compact only for semi-stable, where a is the argument that does not occur in any S stg(f C ). Considering range-maximal (conflict-free or admissible) sets for F we first distinguish between sets S in relation to the argument x. In case x S we have that all arguments from F X are in range, Y is attacked and thus F C needs to be evaluated on its own. In case x S, wlog. assume Y S and a, x 5 S, we have that all of F C and Y are in range, x is attacked and F X needs to be evaluated on its own. This means that either some argument from F C or some argument from F X remains out of range of any semi-stable or stage extension in F and thus stb(f ) =. On a sidenote observe that for very similar reasons F is compact for both, semi-stable and stage semantics. Recall that F C is compact for semi-stable, but not for stage (cf. Theorem 2). This immediately means that for stage semantics there is an implicit conflict between x and F C (argument a to be precise). This also means that for semi-stable semantics there are no implicit conflicts between x and any argument from F C. It remains to show that F indeed is analytic for semi-stable semantics. To this end we still need to investigate possible implicit conflicts between F X and Y, between F C and Y, as well as between F X and F C, and among arguments from F C, as well as among arguments from Y. As mentioned before the range of any semi-stable extension will cover Y and x and either all of F C or all of F X. We start with extensions S with Y S and thus x / S and, wlog. 15

18 fix the evaluation of F X and consider some arbitrary S X sem(f X ). First observe that this immediately means that Y does not contain any conflicts and, due to F X being compact, there are also no conflicts between Y and F X. As Y S X {c, x i } sem(f ) for i {1, 2, 3, 4}, and for i {5, 6, 7} also Y S X {a, x i } sem(f ) as well as Y S X {b, c, x i } sem(f ), there are no conflicts between Y and a, b, c, x 1... x 7, between c and b, x 1... x 7, or between a, b and x 5, x 6, x 7. We now investigate extensions S sem(f ) that contain gradually less arguments from Y. In the following we will omit certain x i from extensions, due to in F C explicit conflicts, for instance x 2 as well as x 4 attack s 1 and t 1. For (Y \ { s 1 } {s 1 }) S we can have x i S for i {1, 3}, and for i {5, 6, 7} on the other hand x i, a S or x i, b S. For (Y \ { t 1 } {t 1 }) S we can have x i, c S for i {1, 3}, or for i {5, 6, 7} on the other hand x i, a S. For (Y \{ū 1 } {u 1 }) S we can have x i, a S or x i, b, c S for i {5, 7}, or for i {1, 2, 3, 4} on the other hand x i, c S. Hence for symmetry reasons for i {1, 2, 3} there are no implicit conflicts between arguments s i, t i, u i on the one side and on the other side Y and arguments a, b, c, x j for j {1, }. Here we can already conclude that there are no implicit but only explicit conflicts between F C and Y in F. For i, j, k {1, 2, 3} fixed and S Y = Y \ { s i, t j, ū k } we have that S X S Y {s i, t j, u k, x i } sem(f ). This means that there are no conflicts between s i, t j and u k, and subsequently that the subframework F C does not have any implicit conflicts in F. Now finally, as elaborated on, each argument from F C can appear in semi-stable extensions S of F that do not contain x and thus contain some arbitrary F X -extension S X. This means that there are no conflicts between F C and F X, which closes the gaps and shows that F indeed is analytic for semi-stable semantics. 3.3 Relations between Compact and Analytic Frameworks In the previous two subsections we have separately investigated relations between semantics for compact and analytic AFs respectively. It looks like the relations (Theorems 2 and 4) are not only similar but indeed equal. The question why we looked at the different classes of AFs separately and whether the equal subset relations are based on stronger similarities must be answered two-fold. On the one hand the examples used for the different proofs share exploitation of similar properties for each semantics considered, and for instance Figure 10 actually builds upon fine-tuned relations between the properties of being compact or analytic. On the other hand in fact not a single example could be used in the other subsection. The compact AFs are not analytic or the analytic AFs are not compact. In what follows we draw some relations between the two classes. We start with similarities as observed in self-loop free AFs. Proposition 5. For any F XAF σ that is self-loop free, F CAF σ (σ {nai, stb, prf, sem, stg}). Proof. Observe that in Definition 3 we allow arguments in conflict to be equal. Hence for any semantics rejected arguments are in conflict with themselves, and rejected arguments in analytic AFs need to be self-attacking. If there is no self-loop in some analytic AF then naturally there is no rejected argument. For naive semantics we can provide even stronger observations. 16

19 Proposition 6. For any self-loop free AF F we have F CAF nai and F XAF nai. Proof. Two self-loop free arguments where none is attacking the other form a conflict-free set. Since we are dealing with finite sets only this immediately means that there is a naive extension containing both arguments. Proposition 7. CAF nai XAF nai. Proof. For an AF F CAF nai it holds that F is self-loop free, hence F XAF nai by Proposition 6. Properness is by the AF ({a}, {(a, a)}), which is nai-analytic, but not nai-compact. However observe that still not every AF is analytic for naive semantics. To see this consider the AF ({a, b}, {(a, a)}). Here {b} is the only naive extension, which means that a and b share an implicit conflict. Finally we conclude this subsection with an observation on the missing relations. That is, we provide reasons why except for naive semantics the properties of being compact or analytic are sufficiently distinct, despite their similarities. Proposition 8. For σ {stb, sem, prf, stg}, we have CAF σ XAF σ and XAF σ CAF σ. Proof. Consider the AF from Figure 2. We have as σ-extensions {a, d} and {b, c}. Hence the AF is compact, but not analytic as the conflict between c and d is implicit only, resulting in CAF σ XAF σ. For XAF σ CAF σ consider the AF ({a, b}, {(a, b), (b, b)}). This AF consists of one accepted and one rejected argument only. It is analytic but not compact. 4 Complexity When aiming for the simplification of an AF along the dimensions of rejected arguments and implicit conflicts the very first questions one has to face is whether there are any rejected arguments or implicit conflicts, in other words whether the AF is already compact, analytic resp., or there is potential for simplifications. That is, in the following we focus on the computational complexity of the following problems for the semantics σ under consideration: (1) decide whether a given AF is σ-compact or not and (2) to decide whether a given AF is σ-analytic or not. Note that the first problem can also be stated as a decision problem for fairness: given an AF, does each argument appear in at least one σ-extension? Further complexity issues for these two classes are mentioned at the end of the section. As being compact means that each argument must be credulously accepted, this question is closely related to credulous reasoning (the decision problem Cred σ is defined by the question whether, given an AF F and an argument a, a is contained in at least one σ-extension of F, i.e. whether a Args σ(f ) holds). Exploiting this observation we first give a generic upper bound for the computational complexity. 17

20 Theorem 9. For any argumentation semantics σ, with Cred σ C for a complexity class C closed under conjunction 4, we have that deciding whether an AF is compact for σ is in C. Proof. By definition an AF F = (A, R) is σ-compact if each a A is credulously accepted w.r.t. σ. Hence to check whether F is compact we simply evaluate a A Cred(F, a), which is only of linear size and by assumption can be evaluated in C as well. We have a similar observation for analytic frameworks, when employing complexity results for Cred 2 σ. Theorem 10. For any argumentation semantics σ, with Cred 2 σ C for a complexity class C closed under conjunction, we have that deciding whether an AF is analytic for σ is in C. Proof. By definition an AF F = (A, R) is σ-analytic if each pair {a, b} A with neither (a, b) R nor (b, a) R is credulously accepted w.r.t. σ. Hence to check whether F is analytic for σ we simply conjoin all these tests (only polynomially many), each of which can be done in C. As P, NP and Σ P 2 are closed under conjunctions we obtain upper bounds for all semantics under our considerations. In particular, we have the following results for naive semantics. Corollary 11. The following problems are in P: 1. Given AF F, deciding whether F CAF nai ; 2. Given AF F, deciding whether F XAF nai. Towards our generic hardness result we introduce the concept of SCC-splittable 5 semantics. Recall that we write F S as shorthand for (A F S, R F (S S)). Definition 5. A semantics σ is called SCC-splittable if there exists a function GF σ : F 2 A 2 A, with F being the set of all AFs over A, such that the following holds for every AF F = (A, R) F. GF σ (F, A) = σ(f ) If A = B C and R does not contain attacks from C to B then σ(f ) = {E E E GF σ (F C\E +, UE C )} F E GF σ (F B,B) with U C E = {c C \ E+ F a B : (a, c) R a E+ F }. 4 A complexity class C is closed under conjunctions iff for any problem Γ C the problem of deciding whether for a finite set of instances of Γ each of these instances is a yes-instance is also in C. 5 Here SCC refer to strongly connected component and reflects the fact that our notion of SCC-splittable is inspired by the notion of SCC-recursiveness [4]. 18

21 Figure 11: The AF F from the reduction in the proof of Theorem 13, for AF F = ({a, b, c, x}, {(a, b), (b, x), (c, x)}). Observe that the second item implies that each strongly connected component of F is either included in B or C. Splitting argumentation frameworks was studied in [6] where (among others) splittings for stable and preferred semantics are presented. Although the splitting theorem in [6] is not stated in terms of Definition 5 it immediately gives a function GF σ with the desired properties. We need one more definition. Definition 6. A semantics σ is called rational, if for any AF F that is a clique (i.e. F is of the form (A, {(a, b) a, b A, a b})) it holds that σ(f ) = {{a} a A F }. Proposition 12. Stable and preferred semantics are rational and SCC-splittable. Next we give the generic hardness results for semantics that are rational and SCC-splittable. Theorem 13. For any rational SCC-splittable argumentation semantics σ deciding whether an AF is compact for σ is as hard as Cred σ when restricted to AFs without self-attacks. Proof. We reduce the problem Cred σ to deciding whether an AF is compact for σ. That is given an instance F = (A, R), x A of Cred σ we build the following AF F = (A A, R R ) with A = {t a a A} and R = {(t a, t b ) a, b A, a b} {(t a, b) a, b A, a x, b a}. That is, we add a clique AF C A = (A, {(t a, t b ) a, b A, a b}) of size A and link it to the original framework as follows: The argument t x does not attack any of the original arguments. All the other arguments t a attack all but one of the original arguments and thus, as we discuss below, enforces that this argument is credulously accepted. The construction is illustrated in Figure 11. To prove the claim we have to show that x is credulously accepted in F iff F is σ-compact. First observe that the new arguments in F form a SCC and are not attacked by arguments from outside. As σ is SCC-splittable we can evaluate F as follows: 1. Compute the extensions of the clique C A. 19